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On spike solutions for a singularly perturbed problem in a compact riemannian manifold
Concentration phenomena for critical fractional Schrödinger systems
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino 'Carlo Bo', Piazza della Repubblica, 13 61029 Urbino (Pesaro e Urbino), Italy |
$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+V(x)u={{Q}_{u}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{u}}(u,v)\ \ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+W(x)v={{Q}_{v}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{v}}(u,v)\ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & u,v>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ in }{{\mathbb{R}}^{N}}, \\ \end{align} & \text{ } & \text{ } & {} \\\end{array} \right.$ |
$\varepsilon>0$ |
$s∈ (0, 1)$ |
$N>2s$ |
$(-Δ)^{s}$ |
$V:\mathbb{R}^{N} \to \mathbb{R}$ |
$W:\mathbb{R}^{N} \to \mathbb{R}$ |
$Q$ |
$K$ |
$C^{2}$ |
$V$ |
$W$ |
References:
[1] |
C. O. Alves,
Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150.
|
[2] |
C. O. Alves, D. C. de Morais Filho and M. A. S. Souto,
On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787.
|
[3] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758.
|
[4] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723.
|
[5] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. |
[6] |
C. O. Alves and S. H. M. Soares,
Existence and concentration of positive solutions for a class
of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457.
|
[7] |
V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator,
J. Math. Phys., 57 (2016), 051502, 18 pp. |
[8] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388). |
[9] |
V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370. |
[10] |
V. Ambrosio,
Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062.
|
[11] |
V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370. |
[12] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645.
|
[13] |
D. Applebaum,
Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics,
116. Cambridge University Press, Cambridge, 2009. xxx+460 pp. |
[14] |
A. I. Ávila and J. Yang,
Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479.
|
[15] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.
|
[16] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
|
[17] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. |
[18] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
|
[19] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
|
[20] |
L. A. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
|
[21] |
W. Choi,
On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153.
|
[22] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235.
|
[23] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
|
[24] |
D. C. de Morais Filho and M. A. S. Souto,
Systems of $p$
-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553.
|
[25] |
M. Del Pino and P. L. Felmer,
Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
|
[26] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[27] |
S. Dipierro, M. Medina, I. Peral and E. Valdinoci,
Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230.
|
[28] |
S. Dipierro, M. Medina and E. Valdinoci,
Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. |
[29] |
S. Dipierro and A. Pinamonti,
A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119.
|
[30] |
L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang,
The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103.
|
[31] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
|
[32] |
G. M. Figueiredo and M. F. Furtado,
Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333.
|
[33] |
A. Fiscella and P. Pucci,
$p$
-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.
|
[34] |
Z. Guo, S. Luo and W. Zou,
On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.
|
[35] |
X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities,
Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp. |
[36] |
H. Hajaiej,
Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704.
|
[37] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
|
[38] |
N. Laskin, Fractional Schrödinger equation,
Phys. Rev. E, 66 (2002), 056108. |
[39] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201.
|
[40] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.
|
[41] |
G. Molica Bisci, V. Rădulescu and R. Servadei,
Variational Methods for Nonlocal Fractional Problems,
with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp. |
[42] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
|
[43] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
J. Math. Phys., 54 (2013), 031501. |
[44] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
|
[45] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
|
[46] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
|
[47] |
E. M. Stein,
Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. |
[48] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
|
[49] |
S. Terracini, G. Verzini and A. Zilio,
Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924.
|
[50] |
Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system,
J. Math. Anal. Appl., 334 (2007), 1308-1325 |
[51] |
K. Wang and J. Wei,
On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153.
|
[52] |
M. Willem,
Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. |
[53] |
Z. Xia, B. Zhang and D. Repovs,
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68.
|
show all references
References:
[1] |
C. O. Alves,
Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150.
|
[2] |
C. O. Alves, D. C. de Morais Filho and M. A. S. Souto,
On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787.
|
[3] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758.
|
[4] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723.
|
[5] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method,
Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. |
[6] |
C. O. Alves and S. H. M. Soares,
Existence and concentration of positive solutions for a class
of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457.
|
[7] |
V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator,
J. Math. Phys., 57 (2016), 051502, 18 pp. |
[8] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388). |
[9] |
V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370. |
[10] |
V. Ambrosio,
Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062.
|
[11] |
V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370. |
[12] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645.
|
[13] |
D. Applebaum,
Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics,
116. Cambridge University Press, Cambridge, 2009. xxx+460 pp. |
[14] |
A. I. Ávila and J. Yang,
Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479.
|
[15] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.
|
[16] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
|
[17] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. |
[18] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.
|
[19] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
|
[20] |
L. A. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
|
[21] |
W. Choi,
On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153.
|
[22] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235.
|
[23] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
|
[24] |
D. C. de Morais Filho and M. A. S. Souto,
Systems of $p$
-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553.
|
[25] |
M. Del Pino and P. L. Felmer,
Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
|
[26] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
[27] |
S. Dipierro, M. Medina, I. Peral and E. Valdinoci,
Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230.
|
[28] |
S. Dipierro, M. Medina and E. Valdinoci,
Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. |
[29] |
S. Dipierro and A. Pinamonti,
A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119.
|
[30] |
L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang,
The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103.
|
[31] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
|
[32] |
G. M. Figueiredo and M. F. Furtado,
Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333.
|
[33] |
A. Fiscella and P. Pucci,
$p$
-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.
|
[34] |
Z. Guo, S. Luo and W. Zou,
On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.
|
[35] |
X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities,
Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp. |
[36] |
H. Hajaiej,
Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704.
|
[37] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
|
[38] |
N. Laskin, Fractional Schrödinger equation,
Phys. Rev. E, 66 (2002), 056108. |
[39] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201.
|
[40] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135.
|
[41] |
G. Molica Bisci, V. Rădulescu and R. Servadei,
Variational Methods for Nonlocal Fractional Problems,
with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp. |
[42] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
|
[43] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
J. Math. Phys., 54 (2013), 031501. |
[44] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
|
[45] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
|
[46] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
|
[47] |
E. M. Stein,
Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. |
[48] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
|
[49] |
S. Terracini, G. Verzini and A. Zilio,
Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924.
|
[50] |
Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system,
J. Math. Anal. Appl., 334 (2007), 1308-1325 |
[51] |
K. Wang and J. Wei,
On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153.
|
[52] |
M. Willem,
Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. |
[53] |
Z. Xia, B. Zhang and D. Repovs,
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68.
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